5. It would take as many hours, as 3 miles are contained in 10 miles. 3 hours and of an hour. 20. They cost as many cents as there are 3 apples in 80 apples; that is, 10 cents. 26. The whole loss was 35 dollars, which was 7 dollars apiece. After this section use the table on page 145. SECTION IV. A. THIS article contains multiplication simply. It is repeating a number a certain number of times and a part of another time. 14. 6 times 5 are 30, and of 5 are 3, which added to 30 make 33. B. In this article the pupil is taught to change a certain number of twos into threes, threes into fives, &c. This article combines all the preceding operations. 24. 4 cords of wood will cost 28 dollars, and 4 of a cord will cost 2 dollars, which makes 30 dollars. 30 dollars will buy 3 hundred-weight of sugar and of another hundredweight. C. 2. The two lemons come to 8 cents, and 8 cents will buy 4 apples, at 2 cents apiece. This is usually called Barter. The general principle is to find what the article will come to, whose price and quantity are given, and then to find how much of the other article that money will buy. 6. If 2 apples cost 4 cents, 1 will cost 2 cents, and 4 will cost 8 cents. 22. Find how many times 2 pears are contained in 20 pears, which is 10 times. 10 times 3 cents are 30 cents. Or, first find what 20 pears would come to at 3 cents piece; and since it is 2 for 3 cents, instead of 1 for 3 cents the price will be half as much. 23. See how many times you can have 5 cents in 30 cents, and you can buy so many times 3 eggs 26. 5 dollars apiece. SECTION V. In this section the principle of fractions is applied to larger numbers, but such as are divisible into the parts proposed to be taken. The pupil, who is familiar with what precedes, will easily understand the examples in this sec tion. They require nothing but division and multiplication. A. Let the pupil explain each example in the following manner. What is 1 sixth of 18? Ans.: 3. Why? Be cause 6 times 3 are 18; therefore if you divide 18 into 6 equal parts, one of the parts will be 3. The pupil will be very likely to say 3 is the 6th part of 18, because 3 times 6 are 18. Be careful to make him say it the other way, viz., 6 times 3 are 18. B. 11. of 20 is 4; 7 are 7 times 4, or 28; and 28 is 4 times 6, and of 6. C. 3. 1 half of 10 is 5, of ten are 4; 5 and 4 are 9. He gave away 9, and had 1 left. 5. 2 yards will cost 1 half of what 4 cost; or 6 dollars. 7. 2 is of 3; therefore 2 oranges will cost of what 3 cost. of 18 cents are 12 cents. 8. of 25 are 20. The ten apples cost 20 cents, which was 2 cents apiece. 13. 3 is of 4. § of 12 dollars are 9 dollars. Or 4 yards at 12 dollars is 3 dollars a yard, and 9 dollars for 3 yards 15. Since 1 is of 3, 7 is † of 3. of 15 cents are 35 cents. Or, oranges at 15 cents, is 5 cents apiece: 7 times 5 cents are 35 cents. Note. In questions of this kind it is generally the simplest way to find wha. 1 article will cost, then it may easily be told how much any number will cost. SECTION VI. A. 4. 2 halves of any number make the whole number Therefore 2 is 1 half of 2 times 2; or 4. It is of 4 times # 2, or 8. Let the pupil answer these questions in the following manner: 4 is of 3 times 4; 3 times 4 are 12. 5 is of ▼ times 5; 7 times 5 are 35. B. 2. 4 is 2 times 2. 16. 2 thirds of any number is twice as much as of the jame number. If 4 is of some number, then 1 half of 4 or 2 is of that number; 2 is of 6; therefore 4 is of 6 23. It is evident that of a pound will cost only of what will cost. If cost 6 cents, will cost 2 cents, and the whole pound 14 cents. 26. It will probably be perceived by this time, that of a number being given, it is necessary to find, and then the number is easily found; 4 being, 2 is, and 2 is of 14. 45. 24 being, of 24 or 3 will be ; 3 is of 27. C. 6. 20 being 4, 5 is 4, and 5 is times 6, and § of 6. D. 4. 18 is 3 times 6, and 6 is Ans. 24 dollars. of 35; and 35 is 5 of 4 times 6, or 24. 6. 54 is of 48; 12 yards at 48 dollars is 4 dollars a yard. He gained 6 dollars. 7. 10 feet is of 15 feet. 8. If water. are under water, there must be out of the 4 is of 12. 10. and are . bear cherries and peaches; consequently, the 10 which bear plums must be the other 4. 10 is of 35. 10 bear peaches, and 15 bear cherries. Miscellaneous Examples. 6. The greyhound gains upon the fox 4 rods in a minute. It will take him 20 minutes to gain 80 rods. 8. of 24. Or you may say, 1 sheep would cost 3 dollars, and 3 sheep 9 dollars. 9. 30 horses will eat 10 times as much as 3 horses. 12. 5 dollars for 1 week, 20 dollars for a month, and 25 dollars for five weeks. 14. It would take them 5 times as long to eat 40 bush els, as it would to eat 8 bushels. 15. 4 horses would eat 4 bushels in 3 days, and it would take them 9 times as long to eat 36 bushels. Ans. 27 days. 16. If 2 men spend 12 dollars in 1 week, 1 man will pend 6 dollars in 1 week, and 30 dollars in 5 weeks, and men would spend 3 times as much, or 90 dollars. 17. The shadow of the staff is of the length of the staff, therefore the shadow of the pole must be the length of the pole. 18 feet is of 27 feet. 20. It would take 2 men 3 times as long to do it as it would 6 men. 23. 8 men would do a piece of work 1 half as large in 2 lays, and it would take 2 men 4 times as long to do it, or 3 days. 28. He must sell it for 56 dollars in order to gain 16 dollars. 56 dollars is 7 dollars per barrel. 29. It cost him 35 dollars, and he must sell it for 45 to gain 10 dollars; 45 dollars is 9 dollars a firkin. 30. Ans. 56 cents. See Section VI. 33. If it would last 3 men 10 months, it would last 1 man 30 months, and 5 men 6 months. 34. There are 8 times 5 in 40; and since the other would build as many times 9 as the first does 5, he would build 8 times 9 or 72 rods. SECTION VII. A. 13. of 20 is 4, are 16; 16 being, 2 is of 14, and 16 is & of 14. is 16. of 28 are 12; 12 is 2 times 6, and 6 is of 48 (12 is of 48), and 48 is 6 times seven and § of 7. B. 1. of 15 are 12; 12 is 6 times 2; 2 is is of 20); of 21 is 7; 18 of 20 (12 of 7. 20 is 2 times 7 and 2. of 18 are 24; 24 is of 27; times 5 and of 5. C. This article contains the multiplication table, in which the numbers from 10 to 20 are multiplied by the ten first numbers. SECTION VIII. A. 1. In one there are 2 halves; in 2 there are twice as many halves, that is 4 halves, or . 2. In one there are 2 halves, in 3 there are three times two halves, that is, 6 halves, or . 3. In one there are 3 thirds, in 2 there are twice 8 hirds, that is, 6 thirds, or f. Each line 15. Draw two lines on the board thus call one; divide the upper line into two equal parts, each part is one half of one, or one half; divide the lower line in the same way, and then call for the answer. The form of the question may be varied by asking how many half apples there are in two apples. 37. In 5 and 2 thirds, how many thirds? Draw five parallel equal lines, divide them into thirds, and draw another parallel with the others and two thirds as long, let the scholar count the thirds. In all these cases of illustration on the board, it is presumed that the scholar will soon take the hint, and devise illustrations for himself at his seat. SECTION IX. A. 2. signifies that 1 thing is divided into 3 equal parts, and 1 part taken. Therefore 2 times 1 third is 2 parts, or . 6. 7 times is, or 21. B. 4. 4 times 2 are 8, and 4 times 1 half are 4 halves, or 2, which added to 8 make 10. 18. 4 times 3 are 12, and 4 times are 12, or three whole ones, which added to 12 make 15. 40. 10 barrels at 3 dollars and a barrel; 10 barrels at 3 dollars, would be 30 dollars, then 10 times § is 50, or 8 and of a dollar. Ans. 38 dollars. C. 2. to each would be 3 times, or, which are 21 Dranges. 3. 1 or 2 bushels. 4. 7 times are 4, or 5 gallons. 5. 8 yards and or two yards, that is 10 yards. 6. 4 times 2 are 8, and 4 times added to 8 make 10 bushels. are 1, or 2, which 12. It would take 1 man 3 times as long as it would 8 Ans. 13 days. men. 14. 3 men would build 3 times as much as 1 man; and n 4 days they would build 4 times as much as in 1 day. Ans. 388 rods. |